{
  "abstract": "The Calderón-Zygmund theory establishes the boundedness of singular integral operators on $L^p$ spaces for $1 < p < \\infty$, yet it encounters a failure at the endpoint $p = 1$. While radial counterexamples in $\\mathbb{R}^n$ are well-documented, Pan-Shao-Wang-Wu \\cite{psww2026} has showed that every nonconstant holomorphic function provides a counterexample to the Poisson equation within the Calderón-Zygmund framework, with the singular locus being a complex subvariety of codimension one. In this paper, we focus on the complex one-dimensional case and establish stronger results. We prove asymptotic formulas with explicit constants for both the level-set integral and the sublevel-set energy. Then we give simplified proofs of the universal counterexamples to Calderón-Zygmund theory at $p = 1$ in $\\mathbb{C}$. Additionally, we construct a new family of counterexamples at the endpoint $p = \\infty$, showing that the failure of $W^{2,\\infty}$-regularity is also a universal phenomenon in complex one dimension.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02601",
  "author_total": 4,
  "authors": [
    {
      "name": "Hongrong Chen"
    },
    {
      "name": "Guokuan Shao"
    },
    {
      "name": "Jujie Wu"
    },
    {
      "name": "Wei Xia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2rz8h5m4evytb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02601",
  "fulltext_url": null,
  "id": "2rz8h5m4evytb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02601",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.02601",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2rz8h5m4evytb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02601",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02601"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Energy asymptotics of holomorphic functions with application to Calderón-Zygmund theory in $\\mathbb{C}$",
  "title_text": "Energy asymptotics of holomorphic functions with application to Calderón-Zygmund theory in ℂ",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/2rz8h5m4evytb",
  "venue": "arXiv",
  "versions": []
}